What Is a Fraction?
A fraction represents a part of a whole. It consists of two numbers separated by a line: the numerator (top number) and the denominator (bottom number). For example, in the fraction ¾, the numerator is 3 and the denominator is 4 — meaning 3 parts out of 4 equal parts.
Types of Fractions
- Proper fractions: The numerator is smaller than the denominator (e.g., 2/5). The value is always less than 1.
- Improper fractions: The numerator is greater than or equal to the denominator (e.g., 7/3). The value is 1 or greater.
- Mixed numbers: A whole number combined with a proper fraction (e.g., 2⅓).
- Equivalent fractions: Different fractions that represent the same value (e.g., 1/2 = 2/4 = 4/8).
How to Add and Subtract Fractions
To add or subtract fractions, you need a common denominator — the same bottom number in both fractions.
- Find the least common denominator (LCD) of the two fractions.
- Convert each fraction so it has the LCD as its denominator.
- Add or subtract the numerators. Keep the denominator the same.
- Simplify the result if possible.
Example: 1/3 + 1/4 → LCD is 12 → 4/12 + 3/12 = 7/12
How to Multiply and Divide Fractions
Multiplying fractions is straightforward: multiply the numerators together, then multiply the denominators together.
Example: 2/3 × 3/5 = 6/15 = 2/5 (simplified)
Dividing fractions uses the "keep, change, flip" method:
- Keep the first fraction as-is.
- Change the division sign to multiplication.
- Flip the second fraction (take its reciprocal).
Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Simplifying Fractions
A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, find the greatest common factor (GCF) of both numbers and divide each by it.
Example: 12/16 → GCF of 12 and 16 is 4 → 12÷4 / 16÷4 = 3/4
Common Mistakes to Avoid
- Adding denominators together instead of finding a common one (e.g., writing 1/2 + 1/3 = 2/5 — this is wrong!).
- Forgetting to simplify the final answer.
- Confusing mixed numbers with improper fractions when calculating.
A Quick Reference Table
| Operation | Rule | Example |
|---|---|---|
| Addition | Find common denominator, add numerators | 1/4 + 1/4 = 2/4 = 1/2 |
| Subtraction | Find common denominator, subtract numerators | 3/4 − 1/4 = 2/4 = 1/2 |
| Multiplication | Multiply top × top, bottom × bottom | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | Keep, change, flip | 1/2 ÷ 1/4 = 1/2 × 4/1 = 2 |
Fractions are foundational to nearly every area of mathematics that follows — from algebra to statistics. Mastering them now builds the confidence needed to tackle more advanced topics.